In order that a relation R defined on a non- empty set A is an equivalence relation, it is sufficient, if R
A)
Is reflexive
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B)
Is symmetric
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C)
Is transitive
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D)
Possesses all the above three properties
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E)
None of these
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The number of reflexive relations of a set with four elements is equal to
A)
\[{{2}^{16}}\]
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B)
\[{{2}^{12}}\]
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C)
\[{{2}^{8}}\]
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D)
\[{{2}^{4}}\]
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E)
None of these
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If \[f(x)=\log \left[ \frac{1+x}{1-x} \right],\] then \[f=\left[ \frac{2x}{1-{{x}^{2}}} \right]\] is equal to
A)
\[{{[f\,\,(x)]}^{2}}\]
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B)
\[{{[f\,\,(x)]}^{3}}\]
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C)
\[2f\,\,(x)\]
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D)
\[3f\,\,(x)\]
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E)
None of these
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The function \[f:R\to R\] defined by \[f\left( x \right)={{e}^{x}}\] is
A)
Onto
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B)
Many-one
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C)
One-One and into
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D)
Many one and onto
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E)
None of these
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If \[{{e}^{x}}=y+\sqrt{1+{{y}^{2}}},\] then y =
A)
\[\frac{{{e}^{x}}+{{e}^{-x}}}{2}\]
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B)
\[\frac{{{e}^{x}}-{{e}^{-x}}}{2}\]
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C)
\[{{e}^{x}}+{{e}^{-x}}\]
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D)
\[{{e}^{x}}-{{e}^{-x}}\]
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E)
None of these
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If \[\sin \left( {{\sin }^{-1}}\frac{1}{5}+{{\cos }^{-1}}x \right)=1,\] then x is equal to
A)
\[1\]
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B)
\[0\]
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C)
\[\frac{4}{5}\]
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D)
\[\frac{1}{5}\]
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E)
None of these
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If \[{{\sin }^{-1}}x+{{\sin }^{-1}}y+{{\sin }^{-1}}z=\frac{\pi }{2},\] then the value of \[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2xyz\] is equal to
A)
0
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B)
1
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C)
2
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D)
3
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E)
None of these
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\[\sin \left\{ {{\tan }^{-1}}\left( \frac{1-{{x}^{2}}}{2x} \right)+{{\cos }^{-1}}\left( \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right) \right\}\] is equal to
A)
\[0\]
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B)
\[1\]
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C)
\[\sqrt{2}\]
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D)
\[\frac{1}{\sqrt{2}}\]
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E)
None of these
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If \[A=\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \\ \end{matrix} \right],\] then \[{{A}^{n}}=\]
A)
\[\left[ \begin{matrix} 1 & n \\ 0 & 1 \\ \end{matrix} \right]\]
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B)
\[\left[ \begin{matrix} n & n \\ 0 & n \\ \end{matrix} \right]\]
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C)
\[\left[ \begin{matrix} n & 1 \\ 0 & n \\ \end{matrix} \right]\]
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D)
\[\left[ \begin{matrix} 1 & 1 \\ 0 & n \\ \end{matrix} \right]\]
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E)
None of these
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If A and B are square matrices of order \[n\times n,\] then \[{{(A-B)}^{2}}\] is equal to
A)
\[{{A}^{2}}-{{B}^{2}}\]
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B)
\[{{A}^{2}}-2AB+{{B}^{2}}\]
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C)
\[{{A}^{2}}+2AB+{{B}^{2}}\]
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D)
\[{{A}^{2}}-AB-BA+{{B}^{2}}\]
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E)
None of these
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Deepak starts walking straight towards east. After walking 75 m he turns to the left and walks 25 m straight. Again he turns to the left and walks a distance of 40 m straight, again he turns to the left and walks a distance of 25 m. How far is he from the starting point?
A)
140 m
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B)
35m
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C)
115 m
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D)
25m
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E)
None of these
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From the following find the correct relation
A)
\[(AB)'=A'B'\]
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B)
\[(AB)'=B'A'\]
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C)
\[{{A}^{-1}}=\frac{adjA}{A}\]
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D)
\[{{(AB)}^{-1}}={{A}^{-1}}{{B}^{-1}}\]
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E)
None of these
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The matrix \[\left[ \begin{matrix} 0 & 5 & -7 \\ -5 & 0 & 11 \\ 7 & 11 & 0 \\ \end{matrix} \right]\] is known as
A)
Upper triangular matrix
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B)
Skew symmetric matrix
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C)
Symmetric matrix
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D)
Diagonal matrix
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E)
None of these
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If \[\left[ \begin{matrix} x+1 & 3 & 5 \\ 2 & x+2 & 5 \\ 1 & 3 & x+4 \\ \end{matrix} \right]=0,\] then x=
A)
\[1,\,\,9\]
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B)
\[-1,\,\,9\]
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C)
\[-1,\,\,-9\]
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D)
\[1,\,\,-9\]
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E)
None of these
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If \[f(x)=\left\{ \begin{matrix} x+\lambda , & x<3 \\ 4, & x=3 \\ 3x-5, & x>3 \\ \end{matrix} \right.\] is continuous at x = 3, then\[\lambda \]=
A)
4
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B)
3
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C)
2
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D)
1
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E)
None of these
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Let \[f(x)=\left\{ \begin{matrix} 0, & x<0 \\ {{x}^{2}} & x\ge 0 \\ \end{matrix} \right.,\] then for all values of x
A)
\[f\]is continuous but not differentiable
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B)
\[f\] is differentiable but not continuous
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C)
\[f'\]is continuous but not differentiable
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D)
\[f'\]is continuous and differentiable
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E)
None of these
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\[\frac{d}{dx}\left[ \cos {{(1-{{x}^{2}})}^{2}} \right]=\]
A)
\[-2x\,\,(1-{{x}^{2}})\,\,\sin \,\,{{(1-{{x}^{2}})}^{2}}\]
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B)
\[-4x\,\,(1-{{x}^{2}})\,\,\sin \,\,{{(1-{{x}^{2}})}^{2}}\]
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C)
\[4x\,\,(1-{{x}^{2}})\,\,\sin \,\,{{(1-{{x}^{2}})}^{2}}\]
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D)
\[-2\,\,(1-{{x}^{2}})\,\,\sin \,\,{{(1-{{x}^{2}})}^{2}}\]
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E)
None of these
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If \[f\,\,(x)={{\tan }^{-1}}\,\,\left( \frac{\sin \,\,x}{1+\cos \,\,x} \right),\] then \[f'\,\,\left( \frac{\pi }{3} \right)=\]
A)
\[\frac{1}{2\,\,(1+\cos x)}\]
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B)
\[\frac{1}{2}\]
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C)
\[\frac{1}{4}\]
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D)
\[\frac{1}{2\,\,{{(1+\sin x)}^{2}}}\]
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E)
None of these
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If \[{{2}^{x}}+{{2}^{y}}={{2}^{x+y}},\] then \[\frac{dy}{dx}=\]
A)
\[{{2}^{x-y}}\frac{{{2}^{y-1}}}{{{2}^{x-1}}}\]
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B)
\[{{2}^{x-y}}\frac{{{2}^{y}}-1}{1-{{2}^{x}}}\]
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C)
\[\frac{{{2}^{x}}+{{2}^{y}}}{{{2}^{x}}-{{2}^{y}}}\]
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D)
\[{{2}^{x+y}}\frac{{{2}^{y}}+1}{{{2}^{x}}+1}\]
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E)
None of these
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The derivative of \[{{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right)\] w.r.t. \[co{{s}^{-1}}\left( \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right)\] is
A)
\[-1\]
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B)
1
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C)
2
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D)
4
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E)
None of these
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If \[Z=si{{n}^{-1}}\left( \frac{x+y}{\sqrt{x}+\sqrt{y}} \right),\] then \[x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}\] is equal to
A)
\[\frac{1}{2}\sin Z\]
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B)
\[\frac{1}{2}\tan Z\]
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C)
0
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D)
\[\frac{1}{2}\]
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E)
None of these
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The angle of intersection of the curves\[y={{x}^{2}}\]and \[x={{y}^{2}}\] at (1, 1) is
A)
\[{{\tan }^{-1}}\left( \frac{4}{3} \right)\]
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B)
\[{{\tan }^{-1}}(1)\]
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C)
\[{{90}^{{}^\circ }}\]
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D)
\[{{\tan }^{-1}}\frac{3}{4}\]
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E)
None of these
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Divide 20 into two parts such that the product of one part and the cube of the other is maximum. The two parts are
A)
(10, 10)
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B)
(5, 15)
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C)
(13, 7)
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D)
(17, 3)
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E)
None of these
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\[\int{\frac{dx}{\sin x+\cos x}=}\]
A)
\[\log \,\,\tan \left( \frac{\pi }{8}+\frac{x}{2} \right)+c\]
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B)
\[\log \,\,\tan \left( \frac{\pi }{8}-\frac{x}{2} \right)+c\]
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C)
\[\frac{1}{\sqrt{2}}\log \,\,\tan \left( \frac{\pi }{8}+\frac{x}{2} \right)+c\]
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D)
\[\frac{1}{2}\log \,\,\tan \left( \frac{\pi }{4}+\frac{x}{2} \right)+c\]
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E)
None of these
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If \[\int{x{{e}^{2x}}}\,\,dx\]is equal to \[{{e}^{2x}}f(x)+C\]where C is constant of integration, then f(x) is
A)
\[\left( 3x-1 \right)\text{/}4\]
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B)
\[\left( 2x+1 \right)\text{/}2\]
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C)
\[\left( 2x-1 \right)\text{/}4\]
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D)
\[\left( x-4 \right)\text{/}6\]
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E)
None of these
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\[\int_{0}^{\pi /2}{\log \tan \,x\,\,dx}=\]
A)
\[\frac{\pi }{2}\,\,{{\log }_{e}}2\]
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B)
\[\frac{-\pi }{2}\,\,{{\log }_{e}}2\]
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C)
\[\pi {{\log }_{e}}2\]
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D)
0
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E)
None of these
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The area bounded by the circle \[{{x}^{2}}+{{y}^{2}}=4,\] line \[x=\sqrt{3}y\] and x - axis lying in the first quadrant, is
A)
\[\frac{\pi }{2}\]
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B)
\[\frac{\pi }{4}\]
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C)
\[\frac{\pi }{3}\]
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D)
\[\pi \]
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E)
None of these
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Which of the following differential equations has the same order and degree?
A)
\[\frac{{{d}^{4}}y}{d{{x}^{4}}}+8\,\,{{\left( \frac{dy}{dx} \right)}^{6}}+5y={{e}^{x}}\]
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B)
\[5\,\,{{\left( \frac{{{d}^{3}}y}{d{{x}^{3}}} \right)}^{4}}+8\,\,{{\left( 1+\frac{dy}{dx} \right)}^{2}}+5y={{x}^{8}}\]
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C)
\[{{\left[ {{\left( 1+\frac{dy}{dx} \right)}^{3}} \right]}^{2/3}}=4\,\,\frac{{{d}^{3}}y}{d{{x}^{3}}}\]
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D)
\[y={{x}^{2}}\frac{dy}{dx}+\sqrt{1+{{\left( \frac{dy}{dx} \right)}^{2}}}\]
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E)
None of these
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The position vectors of the points A, B, C are \[(2i+j\text{ }-k),\] \[(3i-2j\text{ }+k)\] and \[(i+4j-3k)\] respectively. These points
A)
Form an is isoceles triangle
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B)
Form a right-angled triangle
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C)
Are collinear
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D)
Form a scalene triangle
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E)
None of these
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\[\overrightarrow{AB}\times \overrightarrow{AC}=2\hat{i}-4\hat{j}+4\hat{k}.\] Then the area of \[\Delta \,\,ABC\]is
A)
3
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B)
4
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C)
9
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D)
16
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E)
None of these
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If \[\alpha ,\]\[\beta ,\]\[\gamma \] be the angles which line makes with the co-ordinate axes, then
A)
\[{{\sin }^{2}}\alpha +{{\cos }^{2}}\beta +{{\sin }^{2}}\gamma =1\]
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B)
\[{{\cos }^{2}}\alpha +{{\cos }^{2}}\beta +{{\cos }^{2}}\gamma =1\]
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C)
\[{{\sin }^{2}}\alpha +{{\sin }^{2}}\beta +{{\sin }^{2}}\gamma =1\]
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D)
\[{{\cos }^{2}}\alpha +{{\cos }^{2}}\beta +{{\sin }^{2}}\gamma =1\]
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E)
None of these
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The ratio in which the line joining the points \[(2,\,\,4,\,\,5)\] and \[(3,\,\,5,\,\,-4)\] is divided by the yz- plane is
A)
\[2:3\]
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B)
\[3:2\]
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C)
\[-2:3\]
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D)
\[4:-3\]
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E)
None of these
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Two cards are drawn one by one from a pack of cards. The probability of getting first card an ace and second a coloured one is (before drawing second card, first card is not placed again in the pack)
A)
\[\frac{1}{26}\]
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B)
\[\frac{5}{52}\]
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C)
\[\frac{5}{221}\]
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D)
\[\frac{4}{13}\]
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E)
None of these
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In a box of 10 electric bulbs, two are defective. Two bulbs are selected at random one after the other from the box. The first bulb after selection being put back in the box before making the second selection. The probability that both the bulbs are without defect is
A)
\[\frac{9}{25}\]
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B)
\[\frac{16}{25}\]
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C)
\[\frac{4}{5}\]
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D)
\[\frac{8}{25}\]
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E)
None of these
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In a city 20% persons read English newspaper 40% read Hindi newspaper and 5% read both newspapers. The percentage of non-reader of either paper is
A)
60%
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B)
35%
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C)
25%
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D)
45%
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E)
None of these
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Direction: In a given number series after the series below it in the next line a number is given and is followed by (A), (B), (C), (D), (E). You have to complete the series starting with the given number following the pattern of the given series.
140 68 36 16 10 3 284 (A) (B) (C) (D) (E)
Which number will come in the place of (B)?
A)
38
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B)
72
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C)
84
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D)
91
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E)
None of these
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A woman walking with a boy meets another woman and on being asked about her relationship with the boy, she says, "My maternal uncle and his maternal uncle's maternal uncle are brother". How is the boy related to the woman?
A)
Nephew
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B)
Son
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C)
Brother-in-law
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D)
Grandson
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E)
None of these
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Find the missing character from among the given alternatives.
A)
M
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B)
P
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C)
Q
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D)
S
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E)
None of these
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In a dinner party both fish and meat were served. Some took only fish and some only meat. There were some vegetarians who did not take either. The rest accepted both fish and meat. Which of the following logic diagrams correctly reflects this situation?
A)
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B)
done
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C)
done
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D)
done
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E)
None of these
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In a class of 60, where girls are twice that of boys, Kamal ranked seventeenth from the top. If there are 9 girls ahead of Kamal, how many boys are after him in the rank?
A)
3
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B)
7
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C)
12
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D)
23
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E)
None of these
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